Can we achieve what a CNN can do with just a normal neural network?
The convolutional aspect of a CNN comes purely from the connections between layers. Instead of a fully-connected network, which can be difficult to train and tends to overfit more, the convolutional network utilizes hierarchical patterns in the data to limit the number of connections - a local edge detection feature in an image analysis network, for example, only needs input from a small number of local pixels, not the entire image. But in principle, you could assign weights to a fully-connected network to perfectly mimic a convolutional one - you just set the weights of the unneeded connections to zero. Because a general ANN has all the connections present in a CNN plus more, it can do anything a CNN can do plus more, although the training can be more difficult.